Theorem lspsolv 21133

Description: If 𝑋 is in the span of 𝐴 ∪ {𝑌} but not 𝐴, then 𝑌 is in the span of 𝐴 ∪ {𝑋}. (Contributed by Mario Carneiro, 25-Jun-2014.)

Hypotheses

Ref	Expression
lspsolv.v	⊢ 𝑉 = (Base‘𝑊)
lspsolv.s	⊢ 𝑆 = (LSubSp‘𝑊)
lspsolv.n	⊢ 𝑁 = (LSpan‘𝑊)

Assertion

Ref	Expression
lspsolv	⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))

Proof of Theorem lspsolv

Dummy variables 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspsolv.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
2		lspsolv.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lspsolv.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑊)
4		eqid 2737	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2737	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2737	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
7		eqid 2737	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
8		eqid 2737	. . 3 ⊢ {𝑧 ∈ 𝑉 ∣ ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑧(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)} = {𝑧 ∈ 𝑉 ∣ ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑧(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)}
9		lveclmod 21093	. . . 4 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
10	9	adantr 480	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑊 ∈ LMod)
11		simpr1 1196	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝐴 ⊆ 𝑉)
12		simpr2 1197	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ 𝑉)
13		simpr3 1198	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
14	13	eldifad 3902	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑌})))
15	1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14	lspsolvlem 21132	. 2 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
16	4	lvecdrng 21092	. . . . . . 7 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing)
17	16	ad2antrr 727	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (Scalar‘𝑊) ∈ DivRing)
18		simprl 771	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
19	10	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑊 ∈ LMod)
20	12	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑌 ∈ 𝑉)
21		eqid 2737	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
22		eqid 2737	. . . . . . . . . . . . 13 ⊢ (0g‘𝑊) = (0g‘𝑊)
23	1, 4, 7, 21, 22	lmod0vs 20881	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = (0g‘𝑊))
24	19, 20, 23	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = (0g‘𝑊))
25	24	oveq2d 7376	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) = (𝑋(+g‘𝑊)(0g‘𝑊)))
26	11	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝐴 ⊆ 𝑉)
27	20	snssd 4753	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → {𝑌} ⊆ 𝑉)
28	26, 27	unssd 4133	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑌}) ⊆ 𝑉)
29	1, 3	lspssv 20969	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑌}) ⊆ 𝑉) → (𝑁‘(𝐴 ∪ {𝑌})) ⊆ 𝑉)
30	19, 28, 29	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘(𝐴 ∪ {𝑌})) ⊆ 𝑉)
31	30	ssdifssd 4088	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)) ⊆ 𝑉)
32	13	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
33	31, 32	sseldd 3923	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ 𝑉)
34	1, 6, 22	lmod0vrid 20879	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝑊)(0g‘𝑊)) = 𝑋)
35	19, 33, 34	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(0g‘𝑊)) = 𝑋)
36	25, 35	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) = 𝑋)
37	36, 32	eqeltrd 2837	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
38	37	eldifbd 3903	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ¬ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
39		simprr 773	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
40		oveq1 7367	. . . . . . . . . . 11 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑟( ·𝑠 ‘𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
41	40	oveq2d 7376	. . . . . . . . . 10 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) = (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)))
42	41	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → ((𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴) ↔ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)))
43	39, 42	syl5ibcom 245	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)))
44	43	necon3bd 2947	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (¬ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴) → 𝑟 ≠ (0g‘(Scalar‘𝑊))))
45	38, 44	mpd 15	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑟 ≠ (0g‘(Scalar‘𝑊)))
46		eqid 2737	. . . . . . 7 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
47		eqid 2737	. . . . . . 7 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
48		eqid 2737	. . . . . . 7 ⊢ (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
49	5, 21, 46, 47, 48	drnginvrl 20724	. . . . . 6 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ≠ (0g‘(Scalar‘𝑊))) → (((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟) = (1r‘(Scalar‘𝑊)))
50	17, 18, 45, 49	syl3anc 1374	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟) = (1r‘(Scalar‘𝑊)))
51	50	oveq1d 7375	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
52	5, 21, 48	drnginvrcl 20721	. . . . . 6 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)))
53	17, 18, 45, 52	syl3anc 1374	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)))
54	1, 4, 7, 5, 46	lmodvsass 20873	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉)) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
55	19, 53, 18, 20, 54	syl13anc 1375	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
56	1, 4, 7, 47	lmodvs1 20876	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
57	19, 20, 56	syl2anc 585	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
58	51, 55, 57	3eqtr3d 2780	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) = 𝑌)
59	33	snssd 4753	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → {𝑋} ⊆ 𝑉)
60	26, 59	unssd 4133	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑋}) ⊆ 𝑉)
61	1, 2, 3	lspcl 20962	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉) → (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆)
62	19, 60, 61	syl2anc 585	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆)
63	1, 4, 7, 5	lmodvscl 20864	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
64	19, 18, 20, 63	syl3anc 1374	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
65		eqid 2737	. . . . . . 7 ⊢ (-g‘𝑊) = (-g‘𝑊)
66	1, 6, 65	lmodvpncan 20901	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) = (𝑟( ·𝑠 ‘𝑊)𝑌))
67	19, 64, 33, 66	syl3anc 1374	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) = (𝑟( ·𝑠 ‘𝑊)𝑌))
68	1, 6	lmodcom 20894	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) = (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
69	19, 64, 33, 68	syl3anc 1374	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) = (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
70		ssun1 4119	. . . . . . . . . 10 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑋})
71	70	a1i 11	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝐴 ⊆ (𝐴 ∪ {𝑋}))
72	1, 3	lspss 20970	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑋})) → (𝑁‘𝐴) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
73	19, 60, 71, 72	syl3anc 1374	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘𝐴) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
74	73, 39	sseldd 3923	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
75	69, 74	eqeltrd 2837	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
76	1, 3	lspssid 20971	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉) → (𝐴 ∪ {𝑋}) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
77	19, 60, 76	syl2anc 585	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑋}) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
78		snidg 4605	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
79		elun2 4124	. . . . . . . 8 ⊢ (𝑋 ∈ {𝑋} → 𝑋 ∈ (𝐴 ∪ {𝑋}))
80	33, 78, 79	3syl 18	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ (𝐴 ∪ {𝑋}))
81	77, 80	sseldd 3923	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑋})))
82	65, 2	lssvsubcl 20930	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆) ∧ (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})) ∧ 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑋})))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
83	19, 62, 75, 81, 82	syl22anc 839	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
84	67, 83	eqeltrrd 2838	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
85	4, 7, 5, 2	lssvscl 20941	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆) ∧ (((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘(𝐴 ∪ {𝑋})))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
86	19, 62, 53, 84, 85	syl22anc 839	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
87	58, 86	eqeltrrd 2838	. 2 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))
88	15, 87	rexlimddv 3145	1 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  {crab 3390   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890  {csn 4568  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  +_gcplusg 17211  ._rcmulr 17212  Scalarcsca 17214   ·_𝑠 cvsca 17215  0_gc0g 17393  -_gcsg 18902  1_rcur 20153  inv_rcinvr 20358  DivRingcdr 20697  LModclmod 20846  LSubSpclss 20917  LSpanclspn 20957  LVecclvec 21089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-tpos 8169  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-minusg 18904  df-sbg 18905  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-oppr 20308  df-dvdsr 20328  df-unit 20329  df-invr 20359  df-drng 20699  df-lmod 20848  df-lss 20918  df-lsp 20958  df-lvec 21090

This theorem is referenced by:  lssacsex  21134  lspsnat  21135  lsppratlem1  21137  lsppratlem3  21139  lsppratlem4  21140  lbsextlem4  21151  lindsadd  37948  lindsenlbs  37950